Theorem cnmpt1vsca 18184

Description: Continuity of scalar multiplication; analogue of cnmpt12f 17659 which cannot be used directly because .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
tlmtrg.f	|- F = (Scalar ` W )
cnmpt1vsca.t	|- .x. = ( .s ` W )
cnmpt1vsca.j	|- J = ( TopOpen ` W )
cnmpt1vsca.k	|- K = ( TopOpen ` F )
cnmpt1vsca.w	|- ( ph -> W e. TopMod )
cnmpt1vsca.l	|- ( ph -> L e. (TopOn ` X ) )
cnmpt1vsca.a	|- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
cnmpt1vsca.b	|- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )

Assertion

Ref	Expression
cnmpt1vsca	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Distinct variable groups:   x, F   x, J   x, K   x, L   ph, x   x, W   x, X

Allowed substitution hints:   A( x)   B( x)   .x. ( x)

Proof of Theorem cnmpt1vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . . 7 |- ( ph -> L e. (TopOn ` X ) )
2		cnmpt1vsca.w	. . . . . . . . 9 |- ( ph -> W e. TopMod )
3		tlmtrg.f	. . . . . . . . . 10 |- F = (Scalar ` W )
4	3	tlmscatps 18181	. . . . . . . . 9 |- ( W e. TopMod -> F e. TopSp )
5	2, 4	syl 16	. . . . . . . 8 |- ( ph -> F e. TopSp )
6		eqid 2412	. . . . . . . . 9 |- ( Base ` F ) = ( Base ` F )
7		cnmpt1vsca.k	. . . . . . . . 9 |- K = ( TopOpen ` F )
8	6, 7	istps 16964	. . . . . . . 8 |- ( F e. TopSp <-> K e. (TopOn ` ( Base ` F ) ) )
9	5, 8	sylib 189	. . . . . . 7 |- ( ph -> K e. (TopOn ` ( Base ` F ) ) )
10		cnmpt1vsca.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
11		cnf2 17275	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ K e. (TopOn ` ( Base ` F ) ) /\ ( x e. X |-> A ) e. ( L Cn K ) ) -> ( x e. X |-> A ) : X --> ( Base ` F ) )
12	1, 9, 10, 11	syl3anc 1184	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> ( Base ` F ) )
13		eqid 2412	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
14	13	fmpt 5857	. . . . . 6 |- ( A. x e. X A e. ( Base ` F ) <-> ( x e. X |-> A ) : X --> ( Base ` F ) )
15	12, 14	sylibr 204	. . . . 5 |- ( ph -> A. x e. X A e. ( Base ` F ) )
16	15	r19.21bi 2772	. . . 4 |- ( ( ph /\ x e. X ) -> A e. ( Base ` F ) )
17		tlmtps 18178	. . . . . . . . 9 |- ( W e. TopMod -> W e. TopSp )
18	2, 17	syl 16	. . . . . . . 8 |- ( ph -> W e. TopSp )
19		eqid 2412	. . . . . . . . 9 |- ( Base ` W ) = ( Base ` W )
20		cnmpt1vsca.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
21	19, 20	istps 16964	. . . . . . . 8 |- ( W e. TopSp <-> J e. (TopOn ` ( Base ` W ) ) )
22	18, 21	sylib 189	. . . . . . 7 |- ( ph -> J e. (TopOn ` ( Base ` W ) ) )
23		cnmpt1vsca.b	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
24		cnf2 17275	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ J e. (TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( L Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
25	1, 22, 23, 24	syl3anc 1184	. . . . . 6 |- ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
26		eqid 2412	. . . . . . 7 |- ( x e. X |-> B ) = ( x e. X |-> B )
27	26	fmpt 5857	. . . . . 6 |- ( A. x e. X B e. ( Base ` W ) <-> ( x e. X |-> B ) : X --> ( Base ` W ) )
28	25, 27	sylibr 204	. . . . 5 |- ( ph -> A. x e. X B e. ( Base ` W ) )
29	28	r19.21bi 2772	. . . 4 |- ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
30		eqid 2412	. . . . 5 |- ( .s f ` W ) = ( .s f ` W )
31		cnmpt1vsca.t	. . . . 5 |- .x. = ( .s ` W )
32	19, 3, 6, 30, 31	scafval 15932	. . . 4 |- ( ( A e. ( Base ` F ) /\ B e. ( Base ` W ) ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
33	16, 29, 32	syl2anc 643	. . 3 |- ( ( ph /\ x e. X ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
34	33	mpteq2dva 4263	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) = ( x e. X |-> ( A .x. B ) ) )
35	30, 20, 3, 7	vscacn 18176	. . . 4 |- ( W e. TopMod -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
36	2, 35	syl 16	. . 3 |- ( ph -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
37	1, 10, 23, 36	cnmpt12f 17659	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) e. ( L Cn J ) )
38	34, 37	eqeltrrd 2487	1 |- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2674   e. cmpt 4234   -->wf 5417   ` cfv 5421  (class class class)co 6048   Basecbs 13432  Scalarcsca 13495   .scvsca 13496   TopOpenctopn 13612   .s fcscaf 15914  TopOnctopon 16922   TopSpctps 16924   Cn ccn 17250   tX ctx 17553  TopModctlm 18148

This theorem is referenced by:  tlmtgp  18186

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-map 6987  df-slot 13436  df-base 13437  df-topgen 13630  df-scaf 15916  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cn 17253  df-tx 17555  df-tmd 18063  df-tgp 18064  df-trg 18150  df-tlm 18152